Economics 412-1
Dynamic Methods for Economics
Fall 2024

Instructor:

Bruno Strulovici
KGH 3225b-strulovici@northwestern.edu
Office hours: by appointment

Lectures:

Tuesdays and Thursdays from 1:30pm to 3:20pm
Room: KGH 3301

Course Description:
The purpose of the class is to provide students with as many tools and insights as possible to apply optimal control and stopping
theories to economic problems. The class does not replace a rigorous introduction to stochastic processes or optimal control. The
rhythm of the class will be determined by registered students.
Evaluation is based on 3 problem sets and either a term paper (project or review) or the presentation of one paper of interest tothe student either on the theory of optimal control or on some economic application.
Lecture notes will be provided. Reading material and problem sets will be posted on Canvas.
Recommended TextbooksØksendal Stochastic Differential Equations, Springer
Duffie, Dynamic Asset Pricing Theory, Princeton University Press (self-contained appendix)
More advanced:
Karatzas and Shreve, Brownian Motion and Stochastic Calculus, Springer
Revuz and Yor. Continuous Martingales and Brownian Motion, Springer

Deterministic optimal control
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Overview of the Course

Calculus of variation
Euler equation
Maximum principle
Dynamic programming and the Bellman equation
Equivalent formulations (end value vs. flow)
Necessary and sufficient conditions for optimality of solution to Bellman equationo Blackwell criterion for existence and uniqueness of a solution and contraction mapping theorem
o Boundedness of value function
Properties of value function and optimal control
Operator inheritance
Benveniste-Scheinkman theorem

Stochastic processes in continuous time••

Stochastic processes
o Filtration, adapted stochastic process
o Brownian motion
Kolmogorov’s continuity theoremProperties: scaling, volatility identificationo Martingales, submartingales, supermartingales, local martingaleso Doob Meyer decomposition theoremo Quadratic variationo Lévy’s characterization theorem
Stochastic integralo Constructiono Properties: martingale, isometryo Dudley’s theoremo Martingale Representation theoremo Malliavin interpretation•

Itô processes
o Definitiono Markov propertyo Itô’s chain-rule formulao Multidimensional versiono Girsanov’s theorem

Stochastic Differential Equations and Hamilton Jacobi Bellman equation••••

Definition of an SDE, strong and weak solutionso Existence and uniqueness of a strong solutiono Examples: Geometric Brownian motion, Ornstein-Uhlenbeck processes, Bessel processes
Feynman Kac formula
HJB equation: necessity and sufficiency
o Smoothness of value functiono Verification argumento Transversality conditiono Existence uniqueness of solution to HJB equation.
Application: Merton’s portfolio optimization.

Optimal Stopping••••••••••••

Definition, examples of stopping time
Optional sampling theorem
Dynkin’s formula
Strong Markov property
Laplace transform
HJB equation for combined optimal control and stopping.
Smooth pasting property: necessity and sufficiency, intuition
Super-contact condition
Application to corporate finance and optimal default: Leland (1994)
Entry/exit problems: repeated stopping problems with coupled value functions. Dixit (1987)
Switching models, switching costs
Comparative statics of discounting (Quah and Strulovici, 2013)

Experimentation••••••

Multi-armed bandit problem (discounted)
o Problem statement (discrete time)o Whittle’s approacho Gittins indexo Sprouting bandits
Brownian experimentation (Bolton and Harris, 1999)
o Change of measureo Threshold policy
Exponential bandit (Keller, Rady, and Cripps, 2005)
o Single decision maker: learning and updatingo Multi-player experimentation: graphical analysis
Experimentation with heterogeneous types (Strulovici, 2010)
Correlated bandits
Open questions

Principal-Agent problems in continuous time••

IID shocks
o Dynamic incentive compatibility
Direct approachHJB approacho Characterization of optimal contracto Verification argument
Persistent private informationo Stochastic maximum principleo Verification argument

Sannikov “A Continuous-Time Version of the Principal–Agent Problem”
Williams “Persistent Private Information”
Bloedel, Krishna, and Strulovici “Persistent Private Information Revisited”
Strulovici “Renegotiation-Proof Contracts with Persistent Private Information”

HJB equation with jumps and impulse controls
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Lévy processeso Poisson random measureo Itô-Lévy decomposition
Itô-Lévy processes
o Chain-rule formula for Itô-Lévy processes
Impulse control
Viscosity solutions
HJB equation with impulse control
Application: capital markets mobility with catastrophe risk (Duffie and Strulovici, 2010)

Recent Topics
Information structure and dynamic signal acquisition
Zhong (2019), Morris and Strack (WP 2019)
Decision theory
Relation between timing and accuracy of decisions: Fudenberg, Strack, Strzalecki (AER, 2018), Camboni and Durandard (WP)
Relation between uncertainty and discounting (Gabaix Laibson “Myopia and discounting”)
Time dimension as policy space
Callander (AER, 2011), Garfagnini and Strulovici (Restud, 2016), Bardhi (2020,2021)
Dynamic Persuasion, Approval Mechanisms, and Liability Rules
Escudé and Sinander (2019)McClellan (Econometrica, 2021)
Henry Ottaviani (AER 2019)
Poggi Strulovici (WP 2021)
Dynamic games, higher-order beliefs, and filtering
Cisternas and Kolb (WP, 2019)
Bonatti, Cisternas, Toikka (Restud, 2017)
Stopping games
Boli Xu
Mean field games
Ben Moll and co.
Relation between discrete and continuous time
Kreps (2021), Sadzik Stacchetti (2015)
Directed learning
Ke and Villas-Boas (JET, 2019)
Epidemiologic Model
Xiaoyun Qiu (WinP)
Discrimination
Bardhi, Guo, Strulovici (WP 2020)
Two-Dimensional Theories

Durandard Strulovici (2021)
Bardhi (2021)